S 1 Portraits - Michigan State University · 2020-03-16 · S Overview E eat A E 61 2 2 Rhode...
Transcript of S 1 Portraits - Michigan State University · 2020-03-16 · S Overview E eat A E 61 2 2 Rhode...
S 61 2 2 Rhode Portraits and Classification
Plan Overview of Phase Portraits a
Complex EigenvaluesReal Different Eigenvalues
o
Clearview of those Portraits
Review of solution Formulas for 2 2 SEOL DE
This Fundamental sols of 1 A
with A 2 2 real with eigen pairs
It and a V are i
a A 1 real A olio gonalizable
eat it E e't E
b at x Iip F E I i TE CA diagonalizable
eat aglet E Sin t 5
eat Sin pt Ii t coset
c at X To A do I
E e't Lif E eat
ol A a To A not diagonalizable
I eat it E e TETE
where i CA EI 8 8CA EI in it
Example Sketch o phase portrait C diagram of sols of
I A Ie IEogenvals of A det CA AI 1 I I 22.1 4 1
1 I2 at a tip
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II Eti E ie F
i't I it III Lilieat aglet E Sin t 5
E its got ascent to sinks 10L
H I E i IsaEc
to
L Graph components 242I t I 17 1 ai
na aneX za e Sin t
z phase Portrait C diagram phase space
I e ILA e
co Io Fe as time Saf xincreases
0 1711 D
f j ie x
ii cases f.o.fi's k 7t I4
Length of CH i Il call H HD't EH 2
Variant t C42 Sirs42T
L
qitifsaz.IE s nEacesn L
I C0 fqf bEt Phase space
XzC0 BLA
Each E to E fShan Yu masses
D TIM a
I YI l L X
i1 clockwise
ITRemade 4 CenterAll Solutions are closed curves aroundthe t sow
x t an a
Ice c eel to IIc 4
Complex Eigenvalues
Remark i AX A has complex eigenvalues1 a tip RI
ate fIIIeases0
too.uaCenter
Soc
Sink Spirala
F Source spiral
if t fs
Sec
Eigen pairs of I i X 3Itchy
iii II I S Ntieat aglet E Sin E 5eat Sin pt ie t coff t 5
a est ascent to sinks
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the Portera Xz Phase Space
XIII GET Ct is a EicaoAIaETCH
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if
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co e xiao
II j est L Ct Conger than t Ct
3TILO ol e c Ct shorter than Ct
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I HEFCE H 00 Because so Cx 3
t soo
11 II Ct H 0 Because a o ed so
d too
2 ii Cts Udit veto in the11 call direction of ETH
ETH DUE spins Forever
H calltooof is 00
overview I I E i
Xz 1 Xz 1 Xz 1
t II co pet b
iie
Eceie b B
Fol Fol l
t I l oEistable Unstable
47 E increases II E increases
E B E